Approximate methods for solving amplitude-phase problem for discrete signals

Abstract

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.